$$\begin{gathered} In the diagram below,PA and PB are \tan gent to the circle.The AQ is \\ parallel \tan gent PB.PCQ is a secant to the circle and chord AC produced \\ meets PB at D. \\ Show that △CDP is similar to △PDA \\ Hence show that P{D}^2=AD\times CD \\ Hence,or otherwise,prove thatAD bisects PB. \end{gathered}$$

$$\begin{gathered} A particle of mass m is moving in a straight line under the action of a force F, where: \\ F=\frac{m}{{\mathcal{x}}^3}(6-10\mathcal{x}) \\ \left(i\right) Find v^2 in any position if the particle starts from rest at \mathcal{x}=1 \\ \left(ii\right) At which other position does the particle come to rest? \\ Two light inextensible strings PQ and QR each of length l are attached to a particle of mass m at Q. \\ The other ends P and R are fixed to two points in a vertical line such that P is a dis\tan ce l above R. \\ The particle describes a horizontal circle with cons\tan t angular speed \mathcal{w}. \end{gathered}$$

$$\begin{gathered} \left(i\right) Find the tension in the strings. \\ \left(ii\right) What value must \mathcal{w} be greater than in order for the strings to be tight? \end{gathered}$$

$$\begin{gathered} Find, correct to one decimal place, the angle at which a road must be banked so that a car may round a curve with a radius of 200 metres at 100 km/h without sliding. \\ Use a diagram and appropriate force equations in your solution. \end{gathered}$$

$$\begin{gathered} Suppose n is a positive integer. \\ Show that 1-{\mathcal{x}}^2+{\mathcal{x}}^4-{\mathcal{x}}^6+........+{(-1)}^{n-1}{\mathcal{x}}^{2n-2}=\frac{1-{(-1)}^n{\mathcal{x}}^{2n}}{1+{\mathcal{x}}^2} \\ Hence show that \\ -{\mathcal{x}}^{2n}\le \frac{1}{1+{\mathcal{x}}^2}-(1-{\mathcal{x}}^2+{\mathcal{x}}^4-{\mathcal{x}}^6+........+{(-1)}^{n-1}{\mathcal{x}}^{2n-2})\le {\mathcal{x}}^{2n} \\ By integrating over suitable values of \mathcal{x},deduce that \\ -\frac{1}{2n+1}\le \frac{\mathrm{\pi }}{4}-(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+......+{(-1)}^{n-1}\frac{1}{2n-1})\le \frac{1}{2n+1} \\ Explain why \frac{\mathrm{\pi }}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+........ \end{gathered}$$

$$\begin{gathered} The diagram above shows a solid which has the circle {\mathcal{x}}^2+y^2=9 as its base. \\ The cross-section perpendicular to the \mathcal{x} axis is an equilateral triangle. \\ Calculate the volume of the solid. \\ Given that {\mathcal{x}}^4-6{\mathcal{x}}^3+{\mathcal{x}}^2+4\mathcal{x}-12=0 , has a double root at \mathcal{x}=a, \\ Find the value of a. \\ A sequence is defined such that u_1=1,u_2=1 and u_n=u_{n-1}+u_{n-2} for n\ge 3. \\ Prove by induction that u_n<{\left(\frac{7}{4}\right)}^n for integers n\ge 1. \end{gathered}$$

$$\begin{gathered} On polling day the ratio of the electoral votes in the three available booths A, B and C was 5:4:3 respectively. \\ The percentage of votes for Mr Turnbull in these booths was 30\%, 60\% and 50\% respectively. \\ If ten voters were chosen at random, find the probability that Mr Turnbull gained at least eight votes. \\ Give your answer correct to five decimal places. \end{gathered}$$

$$\begin{gathered} A particle is moving in simple harmonic motion about a fixed point O on a line. \\ At time t seconds, it has displacement \mathcal{x}=2 \cos \left(\pi t\right) metres from O. \\ What is the time taken by the particle to travel the first 100 metres of its motion? \\ \left(A\right) 20 seconds \\ \left(B\right) 25 seconds \\ \left(C\right) 50 seconds \\ \left(D\right) 100 seconds \end{gathered}$$

$$\begin{gathered} A three digit numeral n has \mathcal{x}, \mathcal{y} and Ƶ as its digits when writing from left to right respectively. \\ For example, if n=314, then\mathcal{x}=3, \mathcal{y}=1 and Ƶ=4. \\ Show that if \mathcal{x}+Ƶ=\mathcal{y}, then the number is divisible by 11. \end{gathered}$$

$$\begin{gathered} The vertices of △PQR are represented by the complex numbers {\mathcal{z}}_1,{\mathcal{z}}_3, and {\mathcal{z}}_3 respectively. \\ The △PQR is isosceles and right-angled at Q, as shown in the diagram. \end{gathered}$$

$$\begin{gathered} Which of the following statements is true? \\ \left(A\right){\mathcal{z}}_2-{\mathcal{z}}_1=i({\mathcal{z}}_3-{\mathcal{z}}_2) \\ \left(B\right){\mathcal{z}}_1-{\mathcal{z}}_2=i({\mathcal{z}}_3-{\mathcal{z}}_2) \\ \left(C\right){\mathcal{z}}_2-{\mathcal{z}}_1=i({\mathcal{z}}_1-{\mathcal{z}}_3) \\ \left(D\right){\mathcal{z}}_1-{\mathcal{z}}_2=i({\mathcal{z}}_1-{\mathcal{z}}_3) \end{gathered}$$

$$\begin{gathered} A particle of mass, m, travels with cons\tan t velocity, v, in a horizontal circle of radius, R, \\ centre, C, around a track banked at an angle, \alpha , to the horizontal, as shown in the diagram. \\ There is no tendency for the particle to slip sideways. \end{gathered}$$

$$\begin{gathered} What is the expression for the vertical component of the forces acting on the particle? \\ \left(A\right) N\cos \alpha = mg \\ \left(B\right) N\sin \alpha =\frac{m{v}^2}{R} \\ \left(C\right) N\sin \alpha = mg \\ \left(D\right) N\cos \alpha =\frac{m{v}^2}{R} \end{gathered}$$